Magnetic resonance imaging relaxometry (MRIR) concerns the measurement of relaxation rates and/or relaxation times of spins that were excited by nuclear magnetic resonance (NMR). MRIR is based on the physical aspects of nuclei relaxing to the ground state after being excited by radio frequency (RF) energy associated with, for example, a spin inversion recovery sequence. To generate a “map” of a relaxation rate (e.g., R2=1/T2) or of relaxation times (e.g., T1 (spin-lattice relaxation), T2 (spin-spin relaxation), and spin density (M0) relaxation), at least two magnetic resonance (MR) images are acquired. Conventional relaxometry techniques have faced many challenges including accuracy and processing time.
Orthogonal matching pursuit (OMP) has been used to quantify MRI relaxation parameters including T1, T2, and M0. OMP involves finding a correlation between input signals and comparative signals stored, for example, in a dictionary. Items stored in the dictionary may be referred to as “atoms”. Items in the dictionary may also be referred to as “dictionary entries”.
For MRI relaxometry, the OMP dictionary may store simulated signal evolution curves that are a function of two or more relaxation parameters. For example, a dictionary of possible signal evolution curves may be generated for a specific range of possible T1 and T2 values. OMP has even been used to compare under-sampled images to comparative evolution curves to determine relaxation parameter maps. For OMP analysis of other (e.g., non relaxometry) NMR parameter analysis, the OMP dictionary may store simulated signal evolution curves that are a function of two or more NMR parameters.
However, using OMP for relaxometry and other analysis has faced challenges caused by the homogeneity and similarity of dictionary entries. The homogeneity and similarity is more pronounced when OMP is used for simultaneous relaxometry that is tasked with quantifying multiple relaxation parameters (e.g., T1, T2, M0) at the same time. Conventionally, to the extent that OMP dictionaries have been used in relaxometry, they have included very similar dictionary entries, which has lead to inaccurate relaxation parameter quantification. This problem is exacerbated in simultaneous multi-parameter OMP based relaxometry because dictionary entries become even more overly similar when the dictionary entries represent curves associated with variations in more than one relaxation parameter. These types of entries may prove problematic for distinguishing parameters when relaxation is fast and near the acquisition rate.
MRIR is the process of recovering the spin density spectrum from the time sample of the spin signal for pixels in an MR image. Conventionally, MRIR has involved substantial computation that may have lead to unacceptably long parameter determination times. The long computation times have been associated with conventional mathematical approaches to solving the spin relaxometry inverse problem. In addition to taking a long time, the conventional approaches may have yielded inaccurate results.
Conventional mathematical techniques used in MRIR may have initially considered that when an inversion recovery sequence is applied, the ideal value of a pixel's signal S at a time t is described by:S(t)=ρ(1−2e−xt)
where
ρ=spin density,
x=spin relaxation rate in sec−1, and
t=time in seconds.
Some OMP dictionaries may have included dictionary entries based on this initial idealized representation of a pixel's value.
However, since a pixel may represent several types of tissue, and since tissue types may have their own spin density (ρ) and their own spin relaxation rate (x), the ideal signal may be better described by:
      S    ⁡          (      t      )        =            ∑      j        ⁢                  ⁢                  ρ        j            ⁡              (                  1          -                      2            ⁢                                                  ⁢                          ⅇ                              -                xjt                                                    )            
where j covers the different tissue types in the sample.
Alternatively, the ideal signal may be described using:S(t)=Aj+Bje(−t/Cj) 
where A is a constant,
B is a constant,
t is time, and
C is a single relaxation parameter.
Some OMP dictionaries may have included entries based on this more sophisticated representation of a pixel's value.
Even this representation of the S(t) equation may be unsatisfactory because this representation unnaturally assumes that relaxation rates for spins in tissue are unique and distinct with sharp demarcations. It is more likely that spins exhibiting a range of relaxation rates clustered around a central value(s) will be encountered. Therefore, spin density (ρ) may be more of a continuous function and less of a discrete function. When the continuous function approach is observed, then S(t) may be even more realistically described by:S(t)=∫ρ(x)(1−2e−xt)dt 
This representation would yield an idealized curve like curve 100 that is illustrated in FIG. 1. OMP dictionaries may have been built using this integral representation of S(t). However, due to measurement noise and other factors, an actual input curve may be more like curve 200 that is illustrated in FIG. 2. The idealized curve 100 and the actual curve 200 illustrate that MRIR will involve processing the noisy MRI signal (e.g., curve 200) and deriving the spin spectrum that generated the signal. This is known as an inverse problem because it involves working backwards from an observed S(t) to determine the actual input ρ(x). FIG. 3 illustrates the idealized curve 100 super-imposed on the noisy curve 200. OMP-based MRIR involves picking a curve 100 from an available set of idealized curves given noisy signal 200.
FIG. 4 illustrates one set of idealized curves 400, 410, 420, 430, 440, 450, and 460. These curves may all be a function of a single relaxation parameter or of a pair of relaxation parameters where one relaxation parameter is constant. Similar curves may be associated with other NMR parameters including, but not limited to, chemical shift, off-resonance, flow, perfusion, diffusion, motion, and uptake of biomarkers. FIG. 5 illustrates noisy signal 470 super-imposed on the set of curves 400-460. Conventionally, a best fit approach may have selected curve 420 or curve 430 as best matching noisy signal 470. However, a conventional best fit approach may have chosen curve 410 or 440 based on decisions made for an initial part of the input signal and due to sampling and/or under-sampling issues.
Attempts at solving the inverse problem for MRIR have been described as early as 1982. Recent attempts have produced relaxation parameter maps having pixel-wise parameter values for parameters including, but not limited to, T1, T2, and M0 relaxation. Unfortunately, these pixel-wise values may have been exploited in sub-optimal ways due, for example, to issues associated with performing conventional OMP. One issue with conventional OMP is that dictionary entries may be overly homogenous, particularly in the initial portion of a relaxation curve.
Recent advances in quantitative MRI data acquisition have facilitated simultaneously determining multiple relaxation parameters but have yet to provide satisfactory treatment of the inverse problem. For example, Schmitt, et al., Inversion Recovery TrueFISP: Quantification of T1, T2, and Spin Density, Magn Reson Med 2004, 51:661-667, describe extracting multiple relaxation parameters from a signal time course sampled with a series of TrueFISP images after spin inversion. TrueFISP imaging refers to true fast imaging with steady state precession. IR-TrueFISP refers to inversion recovery true fast imaging with steady state precession. TrueFISP is a coherent technique that uses a balanced gradient waveform. In TrueFISP, image contrast is determined by T2*/T1 properties mostly depending on repetition time (TR). As gradient hardware has improved, minimum TRs have been reduced. Additionally, as field shimming has improved, signal to noise ratio has improved making TrueFISP suitable for whole-body applications, for cardiac imaging, for brain tumor imaging, and for other applications. While this represents a significant advance in simultaneously acquiring pixel-wise values, the inverse problem remains an issue.
Doneva, et al., Compressed sensing reconstruction for magnetic resonance parameter mapping, Magn Reson Med 2010, Volume 64, Issue 4, pages 1114-1120 take one approach to the inverse problem. Doneva describes a dictionary based approach for parameter estimating. Doneva applies a learned dictionary to sparsify data and then uses a model based reconstruction for MR parameter mapping. Doneva identifies that “multiple relaxation components in a heterogeneous voxel can be assessed.” The success of this approach depends heavily on the learned dictionary. However, the Doneva library is limited to the idealized, single relaxation parameter curves because the preparation is specific and constrained by the fact that Doneva ultimately reconstructs an image from the acquired data. This constraint may yield a dictionary with overly homogenous relaxation curves against which input curves are to be fit or matched. Also, this approach may be unsuitable for simultaneous multi-parameter relaxometry where multiple relaxation parameters are determined at the same time.
Thus, even though more and more pixel-wise relaxation parameter data is becoming available, and even though that data is becoming available in ever shorter, more clinically relevant time frames, the data may still not lead to acceptably accurate relaxation parameter maps due to issues with solving the inverse problem. Similar issues may remain with other non-relaxation parameters.